Bayesian Analysis
library(rstan)
library(ggmcmc)
library(bayesplot)
library(bridgesampling)load("data_for_stan.RData")
list_for_stan <- list(nY = nrow(data_for_stan), nS = max(data_for_stan$participant), Subj = data_for_stan$participant, size_diff = data_for_stan$size, num_of_trials = data_for_stan$number_of_fixational_trials, Y = data_for_stan$answer.keys)Проверка гипотезы о связи типа иллюзии и разницы между стимулами (больше разница - больше вероятность контрастной иллюзии)
# компилируем модели, соответствующие двум гипотезам
stanmodelH0 <- stan_model('H0_1.stan', model_name = 'H0')
stanmodelH1 <- stan_model('H1_1.stan', model_name = 'H1')
# сэмплируем "предсказания" каждой из гипотез (моделей), нужно сгенерировать очень много
fit_H0 <- sampling(stanmodelH0, list_for_stan, iter = 20000, warmup = 1000)##
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## Chain 4:print(fit_H0)## Inference for Stan model: H0.
## 4 chains, each with iter=20000; warmup=1000; thin=1;
## post-warmup draws per chain=19000, total post-warmup draws=76000.
##
## mean se_mean sd 2.5% 25% 50% 75% 97.5%
## intercept 0.30 0.00 0.39 -0.41 0.03 0.28 0.55 1.10
## beta_sd 1.22 0.00 0.33 0.72 0.99 1.17 1.40 1.99
## beta[1] -0.09 0.00 0.32 -0.74 -0.31 -0.09 0.12 0.53
## beta[2] 0.42 0.00 0.49 -0.48 0.09 0.39 0.72 1.46
## beta[3] -1.28 0.00 0.63 -2.74 -1.63 -1.20 -0.83 -0.29
## beta[4] -1.88 0.00 0.85 -3.88 -2.32 -1.74 -1.28 -0.63
## beta[5] -0.88 0.00 0.48 -1.94 -1.17 -0.83 -0.54 -0.06
## beta[6] 0.92 0.00 0.63 -0.11 0.48 0.84 1.28 2.36
## beta[7] 0.42 0.00 0.38 -0.29 0.16 0.41 0.66 1.22
## beta[8] 0.35 0.00 0.35 -0.32 0.10 0.33 0.57 1.09
## beta[9] 0.01 0.00 0.39 -0.75 -0.24 0.01 0.27 0.80
## beta[10] -1.63 0.00 0.83 -3.62 -2.07 -1.50 -1.05 -0.40
## beta[11] -1.07 0.00 0.41 -1.99 -1.32 -1.03 -0.78 -0.36
## beta[12] -0.99 0.00 0.45 -1.97 -1.26 -0.95 -0.67 -0.20
## beta[13] -1.16 0.00 0.88 -3.23 -1.63 -1.04 -0.55 0.22
## beta[14] -0.71 0.00 0.37 -1.51 -0.94 -0.69 -0.46 -0.05
## beta[15] -1.84 0.00 0.82 -3.81 -2.26 -1.71 -1.26 -0.63
## beta[16] -0.52 0.00 0.33 -1.21 -0.73 -0.50 -0.29 0.10
## beta[17] 0.46 0.00 0.36 -0.20 0.21 0.44 0.69 1.23
## beta[18] -0.80 0.00 0.70 -2.36 -1.20 -0.73 -0.33 0.39
## beta[19] -1.98 0.00 0.80 -3.88 -2.40 -1.85 -1.41 -0.79
## beta[20] 0.17 0.00 0.34 -0.47 -0.06 0.16 0.39 0.86
## beta[21] -1.15 0.00 0.84 -3.16 -1.59 -1.02 -0.56 0.13
## beta[22] -0.88 0.00 0.39 -1.73 -1.12 -0.85 -0.60 -0.19
## beta[23] 0.63 0.00 0.36 -0.01 0.38 0.60 0.85 1.39
## beta[24] 1.05 0.00 0.49 0.22 0.70 1.00 1.34 2.16
## beta[25] 0.07 0.00 0.47 -0.85 -0.24 0.07 0.37 1.02
## beta[26] -1.62 0.00 0.85 -3.63 -2.07 -1.49 -1.02 -0.32
## lp__ -128.33 0.04 4.80 -138.76 -131.35 -127.95 -124.91 -120.02
## n_eff Rhat
## intercept 21284 1
## beta_sd 19044 1
## beta[1] 44734 1
## beta[2] 69192 1
## beta[3] 40548 1
## beta[4] 31713 1
## beta[5] 42251 1
## beta[6] 64130 1
## beta[7] 50712 1
## beta[8] 54525 1
## beta[9] 54264 1
## beta[10] 33796 1
## beta[11] 41271 1
## beta[12] 41058 1
## beta[13] 42345 1
## beta[14] 44790 1
## beta[15] 32982 1
## beta[16] 44559 1
## beta[17] 55574 1
## beta[18] 49312 1
## beta[19] 31470 1
## beta[20] 58536 1
## beta[21] 38604 1
## beta[22] 40576 1
## beta[23] 53063 1
## beta[24] 60305 1
## beta[25] 62868 1
## beta[26] 35946 1
## lp__ 17381 1
##
## Samples were drawn using NUTS(diag_e) at Fri Jan 4 17:30:10 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).print(fit_H1)## Inference for Stan model: H1.
## 4 chains, each with iter=20000; warmup=1000; thin=1;
## post-warmup draws per chain=19000, total post-warmup draws=76000.
##
## mean se_mean sd 2.5% 25% 50% 75% 97.5%
## intercept 1.14 0.00 0.45 0.27 0.84 1.14 1.45 2.05
## beta_mu -1.04 0.00 0.36 -1.79 -1.27 -1.03 -0.80 -0.38
## beta_sd 1.15 0.00 0.28 0.71 0.95 1.11 1.31 1.81
## beta[1] -0.52 0.00 0.35 -1.22 -0.75 -0.52 -0.29 0.15
## beta[2] -0.10 0.00 0.49 -1.01 -0.42 -0.11 0.21 0.92
## beta[3] -1.83 0.00 0.70 -3.44 -2.24 -1.75 -1.33 -0.69
## beta[4] -2.51 0.01 0.90 -4.59 -3.02 -2.39 -1.87 -1.11
## beta[5] -1.39 0.00 0.53 -2.56 -1.72 -1.34 -1.01 -0.46
## beta[6] 0.36 0.00 0.58 -0.62 -0.04 0.30 0.69 1.69
## beta[7] -0.06 0.00 0.39 -0.78 -0.32 -0.07 0.19 0.73
## beta[8] -0.07 0.00 0.36 -0.75 -0.32 -0.08 0.16 0.67
## beta[9] -0.41 0.00 0.41 -1.22 -0.68 -0.41 -0.14 0.39
## beta[10] -2.22 0.00 0.89 -4.31 -2.71 -2.10 -1.60 -0.85
## beta[11] -1.55 0.00 0.47 -2.56 -1.83 -1.51 -1.22 -0.74
## beta[12] -1.52 0.00 0.51 -2.63 -1.84 -1.49 -1.16 -0.62
## beta[13] -1.77 0.00 0.94 -3.98 -2.30 -1.65 -1.11 -0.28
## beta[14] -1.13 0.00 0.41 -2.01 -1.39 -1.10 -0.85 -0.40
## beta[15] -2.43 0.00 0.87 -4.48 -2.92 -2.31 -1.81 -1.08
## beta[16] -0.92 0.00 0.36 -1.68 -1.15 -0.90 -0.67 -0.25
## beta[17] 0.05 0.00 0.36 -0.62 -0.20 0.03 0.28 0.81
## beta[18] -1.40 0.00 0.76 -3.08 -1.85 -1.32 -0.87 -0.10
## beta[19] -2.56 0.00 0.85 -4.57 -3.03 -2.44 -1.96 -1.25
## beta[20] -0.19 0.00 0.35 -0.86 -0.43 -0.20 0.04 0.51
## beta[21] -1.72 0.00 0.92 -3.86 -2.23 -1.58 -1.07 -0.30
## beta[22] -1.35 0.00 0.44 -2.31 -1.62 -1.31 -1.04 -0.56
## beta[23] 0.21 0.00 0.36 -0.43 -0.03 0.20 0.44 0.96
## beta[24] 0.55 0.00 0.47 -0.25 0.22 0.50 0.82 1.59
## beta[25] -0.42 0.00 0.49 -1.37 -0.74 -0.42 -0.10 0.55
## beta[26] -2.23 0.00 0.90 -4.31 -2.75 -2.12 -1.60 -0.81
## lp__ -124.19 0.04 4.87 -134.72 -127.23 -123.82 -120.70 -115.81
## n_eff Rhat
## intercept 19217 1
## beta_mu 22272 1
## beta_sd 22099 1
## beta[1] 36334 1
## beta[2] 51937 1
## beta[3] 37464 1
## beta[4] 31306 1
## beta[5] 40163 1
## beta[6] 52185 1
## beta[7] 40257 1
## beta[8] 43306 1
## beta[9] 48985 1
## beta[10] 34207 1
## beta[11] 37284 1
## beta[12] 36047 1
## beta[13] 40344 1
## beta[14] 40891 1
## beta[15] 33453 1
## beta[16] 40506 1
## beta[17] 47035 1
## beta[18] 48444 1
## beta[19] 31439 1
## beta[20] 48931 1
## beta[21] 40631 1
## beta[22] 36549 1
## beta[23] 42963 1
## beta[24] 50616 1
## beta[25] 53176 1
## beta[26] 36680 1
## lp__ 17506 1
##
## Samples were drawn using NUTS(diag_e) at Fri Jan 4 17:30:57 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).Диагностика
posterior_H0 <- ggs(fit_H0)
posterior_H1 <- ggs(fit_H1)
ggmcmc(D = posterior_H0, file = NULL, family = 'beta', plot = 'ggs_histogram')## Plotting histograms
## Time taken to generate the report: 18 seconds.ggmcmc(D = posterior_H1, file = NULL, family = 'beta', plot = 'ggs_histogram')## Plotting histograms
## Time taken to generate the report: 14 seconds.ggmcmc(D = posterior_H0, file = NULL, family = 'beta', plot = 'ggs_compare_partial')## Plotting comparison of partial and full chain
## Time taken to generate the report: 29 seconds.ggmcmc(D = posterior_H1, file = NULL, family = 'beta', plot = 'ggs_compare_partial')## Plotting comparison of partial and full chain
## Time taken to generate the report: 30 seconds.ggmcmc(D = posterior_H0, file = NULL, family = 'beta', plot = 'ggs_traceplot')## Plotting traceplots
## Time taken to generate the report: 62 seconds.ggmcmc(D = posterior_H1, file = NULL, family = 'beta', plot = 'ggs_traceplot')## Plotting traceplots
## Time taken to generate the report: 64 seconds.ggmcmc(D = posterior_H0, file = NULL, family = 'beta', plot = 'ggs_autocorrelation')## Plotting autocorrelation plots
## Time taken to generate the report: 43 seconds.ggmcmc(D = posterior_H1, file = NULL, family = 'beta', plot = 'ggs_autocorrelation')## Plotting autocorrelation plots
## Time taken to generate the report: 45 seconds.Байес-фактор
# считаем логарифм правдоподобия имеющихся данных для каждой из гипотез (моделей)
H0_res <- bridge_sampler(fit_H0, silent = TRUE)
H1_res <- bridge_sampler(fit_H1, silent = TRUE)
print(H0_res)## Bridge sampling estimate of the log marginal likelihood: -112.1187
## Estimate obtained in 5 iteration(s) via method "normal".print(H1_res)## Bridge sampling estimate of the log marginal likelihood: -107.2271
## Estimate obtained in 5 iteration(s) via method "normal".# смотрим на оценку возможной ошибки подсчета правдоподобия
error_measures(H0_res)$percentage## [1] "0.417%"error_measures(H1_res)$percentage## [1] "0.400%"Предполагаем, что обе гипотезы одинаково вероятны (имеют одинаковую априорную вероятность)
# считаем Байес-фактор (в пользу альтернативной гипотезы)
BF10 <- bf(H1_res, H0_res)
print(BF10)## Estimated Bayes factor in favor of H1_res over H0_res: 133.16600# считаем Байес-фактор (в пользу нулевой гипотезы)
BF01 <- bf(H0_res, H1_res)
print(BF01)## Estimated Bayes factor in favor of H0_res over H1_res: 0.00751Далее: пробуем разные нулевые и альтернативные гипотезы